The Elusive Part of the Standard-Model Extension Gravitational Sector

In the minimal gravitational sector of the Standard-Model Extension, there is a coefficient whose physical consequences are unknown, and the reason behind this lack of effects is still puzzling. This contribution summarizes several studies where the goal was to find a fundamental explanation of this puzzle. So far, no evidence of such a fundamental explanation has been found, suggesting that this coefficient could actually produce physical effects. Nevertheless, while looking for this fundamental reason, several relevant lessons have been revealed.